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278 X EXISTENCE THEOREMS

We write the proof for continuously differentiable mappings, the modifica-
tions in the other cases being obvious.

We may suppose 0 = 0, & = 0, replacing /by the mapping x -*f(a -f x) — i.
Let M be the kernel of the linear mapping /'(O), which is an (n -p)-dimen-
sional subspace of E, and let N be a (p-dimensional) supplement of M in E ; we
take as a basis of E a system (c^i^n ofn vectors such that c_1? . . ,, c_p forma

basis of N, c_p+l9 . . . , c_n a basis of M, and we write x = ^ <P&)c_s for any

»=i
x e E, the (p_t being linear forms. Ife_l9...₉e_nis the canonical basis of K\ we

denote by x~*G(x) the linear mapping x-+ ]T <piO)^ of E onto the

i-p+l

subspace Kⁿ~^p of K" generated by the e_l of index z > /?.

Let P be the image of E (and of N) by the linear mapping /'(O); it is a
p-dimensional subspace of F, having the elements d_t =/'(0) • c_t (1 ^ / ^p)
as a basis; we take a basis (dj)^^ of F, of which the preceding basis

of P form the first p elements, and we write y = ]£ ^_j(y)d_j for anyy e F, the

j = i
i/^. being linear forms. We denote by y-*H(y) the linear mapping

p
y "-* Z ^j(y)^ej °f F ^onto ^^e subspace K^p of K" generated by the e,. of index

We now consider the mapping x -+ ^(x) = H(f(xJ) 4- <?O) of A into K"
which is continuously differentiable. Moreover, by (8.1.3) and (8.2.1), ^we
have g'(x) • s = H(f'(x) - s) + G(s) for any s e E, hence ^'(0) - c_{ == e_t for
1 < f < n (i.e. ^'(°) is represented by the unit matrix with respect to tie bases
(cj) and (e^). Using (10.2.5), we conclude that there is an open neighborhood
U₀ c A of 0 such that the restriction of g to U₀ is a homeomorphism of TJ₀onto an open neighborhood of 0 in K^w, and that the inverse homeomorphism
g~^l is continuously differentiable in #(U₀). Let r > 0 be such that the ball
|x_£| < r (1 < / < n) is contained in ^(U₀), and let U be the inverse image of
that ball by g, which is an open neighborhood of 0; our mapping t/r will be

the restriction to U of the mapping x -> - g(x).

Up to now we have not used the assumption that the rank of f'(x) is
constant in A; this implies that the image P_x of E by/'(*) has dimension p
for any x e A. Now we may suppose U₀ has been taken small enough so
that g'(x) is a linear bijection of E onto K" for x e U₀ (8.3.2); as Pve have
g'(x) • s = H(f'(x) - s) for s e N, the restriction of f'(x) to N must be a
bijection of that ^-dimensional space onto f_x , and the restriction of H to
P_x a bijection of P_x onto K^p. Denote by L_x the bijection of K^p onto f_x , inverse
of the preceding mapping; we can thus write /'(*) = L_x ° H



	278 X EXISTENCE THEOREMS We write the proof for continuously differentiable mappings, the modifica- tions in the other cases being obvious. We may suppose 0 = 0, & = 0, replacing /by the mapping x -f(a* -f x) — i. Let M be the kernel of the linear mapping /'(O), which is an (n -p)-dimen- sional subspace of E, and let N be a (p-dimensional) supplement of M in E ; we take as a basis of E a system (c^i^n ofn vectors such that c_1? . . ,, c_p forma n basis of N, c_p+l9 . . . , c_n a basis of M, and we write x = ^ <P&)c_s for any »=i x e E, the (p_t being linear forms. Ife_l9...₉e_nis the canonical basis of K\ we n denote by x~G(x)* the linear mapping x-+ ]T <piO)^ of E onto the i-p+l subspace Kⁿ~^p of K" generated by the e_l of index z > /?. Let P be the image of E (and of N) by the linear mapping /'(O); it is a p-dimensional subspace of F, having the elements d_t =/'(0) • c_t (1 ^ / ^p) as a basis; we take a basis (dj)^^ of F, of which the preceding basis

	of P form the first p elements, and we write y = ]£ ^_j(y)d_j for anyy e F, the j = i i/^. being linear forms. We denote by y-H(y)* the linear mapping p y "-* Z ^j(y)^ej °f F ^onto ^^e subspace K^p of K" generated by the e,. of index

	We now consider the mapping x -+ ^(x) = H(f(xJ) 4- <?O) of A into K" which is continuously differentiable. Moreover, by (8.1.3) and (8.2.1), ^we have g'(x) • s = H(f'(x) - s) + G(s) for any s e E, hence ^'(0) - c_{ == e_t for 1 < f < n (i.e. ^'(°) is represented by the unit matrix with respect to tie bases (cj) and (e^). Using (10.2.5), we conclude that there is an open neighborhood U₀ c A of 0 such that the restriction of g to U₀ is a homeomorphism of TJ₀onto an open neighborhood of 0 in K^w, and that the inverse homeomorphism g~^l is continuously differentiable in #(U₀). Let r > 0 be such that the ball \|x_£\| < r (1 < / < n) is contained in ^(U₀), and let U be the inverse image of that ball by g, which is an open neighborhood of 0; our mapping t/r will be the restriction to U of the mapping x -> - g(x). Up to now we have not used the assumption that the rank of f'(x) is constant in A; this implies that the image P_x of E by/'() has dimension p for any x e* A. Now we may suppose U₀ has been taken small enough so that g'(x) is a linear bijection of E onto K" for x e U₀ (8.3.2); as Pve have g'(x) • s = H(f'(x) - s) for s e N, the restriction of f'(x) to N must be a bijection of that ^-dimensional space onto f_x , and the restriction of H to P_x a bijection of P_x onto K^p. Denote by L_x the bijection of K^p onto f_x , inverse of the preceding mapping; we can thus write /'() = L_x* ° H